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Numerical schemes for a model for nonlinear dispersive waves. (English) Zbl 0578.65120

Summary: A description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel. The discussion also includes the results of numerical experiments made with each of the schemes. It is suggested, on the basis of these experiments, that one of the schemes may have (discrete) solitary-wave solutions.

MSC:

65Z05 Applications to the sciences
35Q99 Partial differential equations of mathematical physics and other areas of application
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35L05 Wave equation
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References:

[1] Abdulloev, Kh. O.; Bogolubsky, I. L.; Makhankov, V. G., Phys. Lett. A, 56, 427-428 (1976)
[2] Ablowitz, M. T.; Ladik, J. F., J. Math. Phys., 17, 1011-1018 (1976) · Zbl 0322.42014
[3] Alexander, M. E.; Morris, J. L., J. Comput. Phys., 30, 428-451 (1979) · Zbl 0407.76014
[4] Arnold, D. N.; Douglas, J.; Thomée, V., Math. Comp., 36, No. 153, 53-63 (1981) · Zbl 0466.65062
[5] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Philos. Trans. Roy. Soc. London A, 272, 47-78 (1972) · Zbl 0229.35013
[6] Bona, J. L.; Bryant, P. J., (Proc. Cambridge Philos. Soc., 73 (1973)), 391-405 · Zbl 0261.76007
[7] Bona, J. L.; Dougalis, V. A., J. Math. Anal. Appl., 75, 503-522 (1980) · Zbl 0444.35069
[8] Bona, J. L.; Pritchard, W. G.; Scott, L. R., Phys. Fluids, 23, 438-441 (1980) · Zbl 0425.76019
[9] Bona, J. L.; Pritchard, W. G.; Scott, L. R., Philos. Trans. Roy. Soc. London A, 302, 457-510 (1981) · Zbl 0497.76023
[10] Bona, J. L.; Pritchard, W. G.; Scott, L. R., (Lebovitz, N. R., Proceedings of the AMS-SIAM Conference on Fluid Dynamical Problems in Astrophysics and Geophysics. Proceedings of the AMS-SIAM Conference on Fluid Dynamical Problems in Astrophysics and Geophysics, Lectures in Appl. Math., Vol. 20 (1983), Amer. Math. Soc: Amer. Math. Soc Providence, R. I), 235-267 · Zbl 0515.00025
[12] Eilbeck, J. C.; McGuire, G. R., J. Comput. Phys., 19, 43-57 (1975) · Zbl 0325.65054
[13] Eilbeck, J. C.; McGuire, G. R., J. Comput. Phys., 23, 63-73 (1977) · Zbl 0361.65100
[14] Goldstine, H. A., A History of Numerical Analysis from the 16th Through the 19th Century (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0402.01005
[15] Hirota, R., J. Phys. Soc. Japan, 43, 1424-1433 (1977) · Zbl 1334.39013
[16] Lambert, J. D., Computational Methods in Ordinary Differential Equations (1973), Wiley: Wiley London · Zbl 0258.65069
[17] Peregrine, D. H., J. Fluid Mech., 25, 321-330 (1966)
[18] Santarelli, A. R., Nuovo Cimento B, 46, 179-188 (1978)
[19] Showalter, R. E., Appl. Anal., 7, 297-308 (1978) · Zbl 0387.34043
[20] Stoer, J.; Bulirsch, R., Introduction to Numerical Analysis (1979), Springer-Verlag: Springer-Verlag New York · Zbl 0771.65002
[21] Wahlbin, L., Numer. Math., 23, 289-303 (1975) · Zbl 0283.65052
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